Learning (Approximately) Equivariant Networks via Constrained Optimization

2025-05-19

AI Summary

Real-world data often violate ideal symmetries due to noise and measurement biases, hindering the training of strictly equivariant models; yet abandoning equivariance entirely forfeits valuable structural inductive biases. To address this trade-off, we propose Adaptive Constrained Equivariance (ACE), a framework grounded in homotopy continuation that dynamically relaxes or tightens equivariance constraints during training, enabling data-driven balancing between equivariance and expressivity. ACE integrates differentiable constraint relaxation, Lagrangian multiplier–based optimization, and neural network regularization—thereby circumventing the parameter-space restrictions imposed by hard equivariance constraints. Extensive experiments across diverse architectures and tasks demonstrate that ACE consistently improves model accuracy, sample efficiency, and robustness to input perturbations, outperforming both strictly equivariant baselines and heuristic relaxation approaches.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Machine Learning: Learning with Manifolds
• Computer Vision: Adversarial Attacks & Robustness

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Equivariant neural networks are designed to respect symmetries through their architecture, boosting generalization and sample efficiency when those symmetries are present in the data distribution. Real-world data, however, often departs from perfect symmetry because of noise, structural variation, measurement bias, or other symmetry-breaking effects. Strictly equivariant models may struggle to fit the data, while unconstrained models lack a principled way to leverage partial symmetries. Even when the data is fully symmetric, enforcing equivariance can hurt training by limiting the model to a restricted region of the parameter space. Guided by homotopy principles, where an optimization problem is solved by gradually transforming a simpler problem into a complex one, we introduce Adaptive Constrained Equivariance (ACE), a constrained optimization approach that starts with a flexible, non-equivariant model and gradually reduces its deviation from equivariance. This gradual tightening smooths training early on and settles the model at a data-driven equilibrium, balancing between equivariance and non-equivariance. Across multiple architectures and tasks, our method consistently improves performance metrics, sample efficiency, and robustness to input perturbations compared with strictly equivariant models and heuristic equivariance relaxations.

Problem

• Handling imperfect symmetries in real-world data
• Balancing equivariance and flexibility in neural networks
• Improving model performance without strict symmetry constraints

Innovation

• Adaptive Constrained Equivariance balances symmetry flexibility
• Gradual tightening smoothes training via homotopy principles
• Data-driven equilibrium improves robustness and efficiency